Thread: Convergence in L1 and characteristic function

If for , and in , then f equals to the characteristic function of a measurable set.

Proof so far.

So I know that

And I need to find a measurable set, say M, such that

I apologize for the lack of significant amount of work here, but I just don't really know how to start here, any hints?

Thank you.

Actually , I think , f equals to the characteristic function of a measurable set, a. e.
Hint :convergence in implies convergence in measure, and convergence in measure implies a subsequence convergence a. e.